2x2 Matrices and Differences of Fractions

Question

Consider the difference of two arbitrary fractions, $\frac{a}{b}$ and $\frac{c}{d}$. $$\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}$$ The numerator is the determinant of the 2x2 matrix $$ \left( \begin{array}{ccc} a & c \\ b & d \\ \end{array} \right)$$ Is there any reason for this? Are the two related in any way?

Answer 1

Think of the determinant as an expression for an area or volume spanned by the vectors $(a,b)^T$ and $(c,d)^T$. If the ratios, which represent the direction of the vector are equal, i.e. $a/b=c/d$, then the area/volume is $0$.